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MATHEMATICAL  METHODS  IN  PHYSICS. 


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BY  JAMES  BYRNIE  SHAW. 


Reprinted  from  the 

BULLETIN  OF  THE  AMERICAN  MATHEMATICAL  SOCIETY 
2d  Series,  Vol.  XXI.,  No.  4,  pp.  192-199 
New  York,  January,  1915 


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[Reprinted  from  Bull.  Amer.  Math.  Society,  Vol.  21,  No.  4,  Jan.,  1915.] 


MATHEMATICAL  METHODS  IN  PHYSICS. 

Sur  quelques  Progres  recents  de  la  Physique  mathematique.  Par 
Vito  Volterra,  Clark  University  Lectures  of  1909,  pub- 
lished by  Clark  University,  1912.  82  pp. 

Drei  Vorlesungen  iiher  neuere  Fortschritte  der  mathematischen 
Physik.  Von  Vito  Volterra,  mit  Zusatzen  und  Ergan- 
zungen  des  Verfassers.  Deutsch  von  Dr.  Ernst  Lanela. 
Sonderabdruck  aus  dem  Archiv  der  Mathematik  und  Physik , 
III.  Reihe,  Band  XXII,  Heft  2/3.  B.  G.  Teubner.  Leipzig. 
Lemons  sur  V Integration  des  Equations  differ entielles  aux  Derivees 
partielles.  Par  Vito  Volterra.  Professees  a Stockholm. 
Nouveau  tirage.  Paris,  Hermann.  1912.  3 + iv  + 83  pp. 
The  first  of  these  books  consists  of  three  lectures  delivered 
at  Clark  University,  and  afterwards  printed  by  the  Uni- 
versity.* They  have  since  appeared  in  the  second  form  in 
German  in  the  Archiv  der  Mathematik  und  Physik,  (3),  22 
(1914),  pages  97-182.  In  the  latter  form  some  of  the  details 
omitted  in  the  original  are  supplied.  The  third  book  is  a 
reprint  of  lectures  delivered  at  Stockholm  in  1906.  There 
have  been  added  some  corrections,  and  some  bibliographical 
notes.  These  lectures  are  striking  examples  of  the  intimate 
relationship  between  the  advance  of  mathematics  and  that  of 
physics. 

The  fundamental  notion  of  the  Stockholm  lectures  is  that 
the  theories  of  the  propagation  of  heat,  of  hydrodynamics, 
elasticity,  Newtonian  forces,  and  electromagnetism  can  all 
be  treated  from  a single  point  of  view — reducing  indeed  to 
differential  equations  of  the  same  general  form  but  of  three 
types,  the  facts  and  the  processes  used  following  the  types. 
A good  supplementary  paper  to  read  along  with  the  first  part 
of  the  lectures  on  differential  equations,  containing  examples 
and  more  detail,  is  to  be  found  in  the  Annates  de  VEcole 
Normale,  (3),  24  (1907),  page  411.  The  most  interesting  part  of 
the  lectures  is  the  introduction  of  the  notion,  due  to  Professor 
Volterra,  of  function  of  a line.  In  the  fifth  lecture  this  notion 
appears,  and  is  indeed  the  guide  to  a generalization  of  the 

* The  volume  also  contains  lectures  by  Rutherford : ‘ ' History  of  the 
alpha-rays  from  radio-active  substances”;  Wood:  “The  optical  properties 
of  metallic  vapors”;  Barus:  “Physical  properties  of  the  iron  carbides.” 


MATHEMATICAL  METHODS  IN  PHYSICS. 


193 


[Jan., 


analytic  functions  of  a complex  variable.  The  particular 
function  of  a line  used  in  this  lecture  is  the  line  integral 

V = Wo  + f ( Xdx  + Ydy  +[Zdz), 

where  uQ  is  a constant  and  the  line  integral  extends  from  a 
point  A to  a point  B.  If  we  set  X'  = (dY/dz  — dZ/dy), 
Y'  = (dZ/dx  — dX/dz ),  Z'  = ( dX/dy  — dY/dx),  then,  n being 
the  outward  normal  to  the  surface  enclosed,  the  line  integral 
in  question  around  a loop  will,  by  Stokes’s  theorem,  be  the 
same  as 

W =ff  (. X'  cos  nx  + Y'  cos  ny  + Z'  cos  nz)dA. 

If  we  let  the  loop  decrease  and  determine  the  limit  of  the  ratio 
of  W to  the  area  enclosed,  as  the  vanishing  loop  approaches 
a point,  the  limit  in  question  is  nothing  else  than  the  projec- 
tion, on  the  normal  to  the  surface  at  the  point,  of  the  vector 
whose  components  are  X',  Y',  Z',  that  is,  of  the  curl  of  the 
vector  X,  Y,  Z.  This  projection,  if  found  for  a point  on  the 
path  of  integration,  Professor  Volterra  calls  the  derivative 
of  the  function  of  the  line  V with  respect  to  the  surface, 
and  he  represents  the  curl  by  the  symbols 

X'  = dV/d(yz)  Y'  = dV/d(zx ) Z'  = dV/d(xy ). 

If  now  there  is  a function  u whose  gradient  is  (. X',  Y',  Z'), 
that  is,  if 

du/dx  = X',  du/dy  = Y',  du/dz  = Z', 

then  the  convergence  of  the  gradient  of  u is  zero,  and  u is 
harmonic  since  X72u  = 0.  When  these  relations  are  satisfied 
we  have  u and  V so  related  that  one  may  be  called  the  con- 
jugate of  the  other.  This  is  the  generalization  referred  to 
of  the  theory  of  complex  variables.  An  easy  mathematical 
example  is  found  by  setting 

u = z2  + y2  - 2z2, 

which  is  harmonic,  and  whose  gradient  is  the  vector 
(2x,  2y,  - 4s). 

It  is  conjugate  to  the  function  (integral  around  a loop) 

V = f (2 zydx  — 2 zxdy), 


1915.]  MATHEMATICAL  METHODS  IN  PHYSICS.  194 

since  the  curl  of  (2 zy,  — 2zx,  0)  is  (2x,  2 y,  — 4s),  the  gradient 
of  u.  An  easy  physical  example  is  the  field  of  potential  at  a 
fixed  origin  of  a single  magnet  pole  in  empty  space,  as  the 
pole  is  moved  into  all  possible  positions,  which  gives  the 
function  u;  and  the  field  of  a circuit  carrying  a unit  electric 
current,  as  it  is  moved  into  all  possible  positions  and  shapes, 
which  gives  the  potential  V at  the  fixed  origin.  The  two  func- 
tions are  conjugate,  the  first  a function  of  a point,  the  reciprocal 
of  its  distance  from  the  origin;  the  latter  a function  of  a line, 
the  solid  angle  it  subtends  at  the  origin. 

The  function  of  a line  of  course  need  not  be  conjugate  to  a 
function  u.  In  case  X',  Y',  Z'  is  a vector  whose  convergence 
is  zero,  then  it  may  be  written  as  the  curl  of  a vector  X,  Y,  Z, 
the  well-known  relation  of  a vector  to  its  vector  potential; 
and  the  integral 

//<*'  cos  nx  + Y'  cos  ny  + Z'  cos  nz)dA 

= J (Xdx  + Ydy  + Zdz) 

gives  a function  of  a line.  It  need  be  remarked  that  these 
definitions  of  function  of  a line  and  the  differential  of  a func- 
tion of  a line  are  generalized  somewhat  in  the  calculus  of 
fonctionnelles. 

The  notion  of  monogenicity  is  extended,  under  the  name 
isogenicity,  to  space  of  three  dimensions  in  the  following 
form.  Two  functions  of  a complex  variable  2 are  monogenic 
if  their  differentials  at  any  point  have  a ratio  / which  is  a func- 
tion of  the  point  but  not  of  the  direction  of  the  differen- 
tials. Two  functions  of  a line  are  isogenic  if  at  each  point  of 
the  line  the  vector  derivative  of  each,  that  is,  the  curl  of  the 
vector  which  is  to  be  integrated  along  the  line,  is  parallel  to 
the  vector  derivative  of  the  other.  In  vector  notation  we 
would  write  the  definition 

WB1=fWB2, 

where/ is  independent  of  the  derivative  plane  as  defined  above. 
The  idea  of  function  of  a line  may  evidently  be  extended  to 
functions  of  surfaces  and  hyperspaces,  and  leads  into  the 
calcul  fonctionnel. 

These  functions  of  lines  and  surfaces,  and  for  the  most 
general  case,  of  hyperspaces,  enable  one  to  understand  the 
applibation  of  the  methods  of  Jacobi  and  Hamilton  to  the 


195  MATHEMATICAL  METHODS  IN  PHYSICS.  [Jan., 

problems  of  the  calculus  of  variations.  The  simple  integral 
of  the  ordinary  theory  may  be  considered  as  a function  of  its 
limits  and  of  the  values  of  unknown  functions  at  the  limits. 
The  extensions  to  multiple  integrals  are  easily  suggested  by 
this  view  of  the  procedure,  namely,  we  must  consider  the 
integrals  to  be  functions  of  the  lines,  or  surfaces,  or  hyper- 
spaces, that  bound  the  space,  and  of  the  values  of  unknown 
functions  on  these  contours. 

The  detailed  treatment  of  the  partial  differential  equations 
is  to  be  found  in  the  last  four  lectures,  and  reference  to  them 
is  necessary  to  have  a clear  notion  of  them. 

In  the  course  at  Clark  University  we  find  the  dominant  idea 
again  to  be  the  unifying  principles  of  the  application  of  mathe- 
matics to  mechanics,  elasticity,  and  mathematical  physics. 
The  first  lecture  is  devoted  to  showing  the  reduction  of  physical 
problems  to  problems  in  the  calculus  of  variations,  the  second 
to  the  advance  in  methods  in  elasticity,  and  the  third  to  the 
problem  of  heredity,  which  leads  to  Volterra’s  integro-differ- 
ential  equations.  These  three  lectures  we  will  examine  in 
some  detail. 

In  the  first  there  is  given  a reduction  of  the  problem  of 
electrodynamics  into  terms  of  the  variation  of  the  definite 
integral 

P = fdtif '/s(arsZrX8  + prsLrh)dA, 

in  which  the  quantities  a,  (3  are  quite  arbitrary.  This  varia- 
tion in  terms  of  purely  arbitrary  quantities  shows  us  that  we 
can  devise  an  infinity  of  mechanical  explanations  or  models 
of  electrodynamic  phenomena.  But  this  analytic  form  of 
the  problem  enables  us  to  introduce  curvilinear  coordinates, 
and  thus  consider  curvilinear  spaces.  We  also  are  enabled  to 
find  the  integral  invariants,  and  to  apply  Volterra’s  reciprocity 
theorem  which  corresponds  to  Green’s  theorem,  and  further 
to  introduce  the  generalization  of  the  Hamilton-Jacobi 
methods  and  the  principle  corresponding  to  stationary  action 
and  varying  action.  In  order  to  accomplish  this,  the  functions 
of  lines,  surfaces,  and  hyperspaces  have  to  be  used.  We  thus 
come  into  contact  with  the  theory  of  the  inversion  of  definite 
integrals,  that  is,  the  solution  of  linear  integral  equations, 
and  with  the  study  of  functions  of  variables  which  run  over 
assemblages  of  curves,  of  surfaces,  etc.  We  are  brought  up 
to  the  functional  calculus,  or  as  it  has  been  called,  general 


1915.]  MATHEMATICAL  METHODS  IN  PHYSICS.  196 

analysis.  The  generalization  of  the  Hamilton- Jacobi  method 
replaces  the  canonical  equations  by  partial  derivatives,  and 
the  partial  differential  equation  of  Jacobi  is  replaced  by  a 
functional  equation. 

There  is  an  intimate  connection  between  the  partial  dif- 
ferential equation,  its  characteristics,  and  the  theory  of  waves.* 
It  is  simply  necessary  to  consider  the  variable  t as  on  the  same 
footing  as  the  variables  x,  y , z.  The  bearing  of  this  is  shown 
in  the  closing  considerations  of  the  first  lecture.  For  example, 
in  the  very  simple  case  of  the  equation  of  a vibrating  mem- 
brane 

d2u/dt 2 — d2u/dx 2 — d2u/dy2  = 0 

we  have  the  partial  differential  equation  corresponding  to  the 
vanishing  of  the  variation  of  the  integral 

V = f f f [( du/dt )2  — ( du/dx )2  — (du/dy)2]dxdydt. 

The  surfaces  of  discontinuous  derivatives  and  variation  of  V 
always  equal  to  zero  are  then  the  envelopes  of  the  character- 
istic cones  of  the  partial  differential  equation.  The  general 
question  of  waves  is  therefore  considered,  and  Minkowski’s 
universe  noticed.  This  naturally  leads  to  the  Lorentz  trans- 
formation, and  to  Poincare’s  demonstration  that  under  this 
transformation  the  integral  whose  variation  was  considered, 
and  which  may  be  called  the  action,  remains  invariant. 

In  the  second  lecture  the  development  of  methods  in  the 
theory  of  elasticity  is  taken  up,  particularly  those  that  are 
connected  with  the  ideas  already  mentioned.  The  two  great 
classes  of  methods  of  integrating  the  differential  equations 
of  elasticity  may  be  called  the  method  of  Green  with  its  exten- 
sions, and  the  method  of  simple  solutions.  Green’s  method 
further  has  two  divisions,  in  one  of  which  the  conception  of 
Green  alone  is  sufficient  to  solve  the  problem,  in  the  other  we 
must  add  consideration  of  the  characteristics.  Green’s  method 
starts  with  Laplace’s  equation,  and  depends  upon  a reci- 
procity theorem,  by  means  of  which  from  a fundamental 
solution  one  is  enabled  to  determine  a harmonic  function 
inside  a given  region  when  its  values  on  the  contour  are  given. 
Betti  carried  the  method  of  Green  over  into  elasticity  and 
extended  the  reciprocity  theorem  by  the  proposition:  If  two 

* Encyclopedic  des  Math.,  II4  1 (II,  22,  8). 


197  MATHEMATICAL  METHODS  IN  PHYSICS.  [Jan., 

systems  of  exterior  forces  determine  two  systems  of  displace- 
ments in  an  elastic  body,  the  work  done  by  either  in  producing 
the  displacement  due  to  the  other  is  the  same. 

The  second  division  of  methods  along  the  line  started  by 
Green  is  found  in  Kirchoff’s  work  on  the  equation  of  retarded 
potential  on  four  variables.  He  succeeded  in  solving  it  by 
means  of  a fundamental  solution  due  to  Euler,  but  in  order 
to  perceive  the  real  difficulties  in  the  way  it  is  necessary  to 
consider  the  problem  on  only  three  variables.  The  char- 
acteristics enter  into  the  solution  radically  in  this  case.  The 
lecturer  shows  the  inherent  difference  between  the  cases  of 
three  dimensions  and  four  dimensions.  Again  when  the  body 
is  not  simply  connected  and  the  functions  can  be  polydromic 
the  method  of  Green  needs  further  extensions.  The  reci- 
procity theorem  enters  in  as  a theorem  of  the  symmetry  of  a 
set  of  coefficients  Eij  = Eji,  which  occur  in  the  linear  equations 
that  give  the  efforts  in  terms  of  the  distortions. 

The  method  of  simple  solutions  has  been  given  great  power 
by  the  development  of  the  theory  of  integral  equations,  and 
the  determination  of  methods  of  expansion  in  series  of  ortho- 
gonal functions. 

The  third  lecture  introduces  the  new  developments  due  to 
Professor  Volterra  himself  and  now  well-known.  These  lead 
to  the  division  of  mechanics  into  the  mechanics  of  no  heredity, 
wherein  the  state  of  a system  depends  only  upon  the  infini- 
tesimally near  states  preceding,  and  the  mechanics  of  heredity, 
in  which  the  state  depends  upon  all  the  preceding  states,  thus 
introducing  an  action  at  a time-distance.  A simple  example 
is  used  to  illustrate  the  new  problem,  the  dependence  of 
the  angle  of  torsion  of  a wire  upon  the  moment  of  torsion. 
Instead  of  Hooke’s  law 

co  = KM, 

we  find  it  must  be  expressed  by  a more  elaborate  law  dependent 
upon  the  time 

co  = KM(t ) + r M(t)ip(t,  t)cIt 

*Jt0 

+AC*-jC  dT2M(Ti)M(T2)<p(t,  n,  r2)  + 

In  the  case  of  linear  heredity  this  expression  terminates  with 
the  second  term.  There  arise  now  several  questions. 


1915.] 


MATHEMATICAL  METHODS  IN  PHYSICS. 


198 


(1)  What  is  the  significance  of  the  coefficient  <p(t,  r)? 

(2)  When  is  known,  how  may  M be  found  for  a given  co? 

(3)  How  is  <p  determined  when  it  is  unknown? 

(4)  Is  it  possible  to  extend  these  conceptions  to  the  general 
problem  of  elasticity? 

(5)  Is  it  possible  to  extend  these  conceptions  to  the  phe- 
nomena of  magnetism  and  electricity? 

(6)  What  phenomena  will  be  amenable  to  this  method  of 
treatment? 

For  the  first  three  questions  it  is  found  that  the  methods  of 
integral  equations  are  sufficient  to  furnish  the  answer.  How- 
ever for  the  fourth  a new  type  of  equation  is  in  evidence,  the 
integro-differential  equation.  We  meet  this  indeed  in  the 
problem  of  the  wire  itself  when  we  study  its  oscillations.  The 
integro-differential  equation  involves  the  partial  derivatives 
of  the  unknown  function  as  well  as  integrals  containing  the 
unknown  function.  The  answer  to  the  fifth  question  leads 
also  to  integro-differential  equations  for  the  electromagnetic 
equations. 

To  resolve  equations  of  this  form  a further  extension  of  the 
method  of  Green  is  necessary,  and  a new  reciprocity  theorem 
arises.  Fundamental  solutions  may  be  found  and  from  these 
arises  the  complete  solution. 

In  applying  algebra  to  natural  phenomena  we  have  the 
great  advantage  that  we  can  postpone  to  the  last  moment  the 
specification  of  the  constants  that  enter  the  natural  problem, 
and  are  thus  in  a much  better  position  to  consider  the  related 
hypotheses.  So  too  in  the  functional  calculus  in  all  its  forms, 
we  are  able  to  postpone  the  specification  of  the  functions 
entering  the  problem  until  the  last  moment,  and  are  thus 
able  to  avoid  vicious  hypotheses  in  the  beginning.  Without 
these  developments  many  types  of  problem  become  impossible 
of  solution.  Further,  advances  in  mathematical  physics  come 
from  the  subsumption  under  one  law  of  many  different  classes 
of  phenomena. 

In  the  translation  of  the  Clark  University  Lectures  many 
details  of  the  transformations  in  the  first  lecture  are  supplied, 
to  the  relief  of  the  general  reader.  Further,  the  references 
have  been  made  more  precise  and  have  been  inserted  on  the 
pages  as  footnotes  instead  of  being  collected  at  the  ends  of 
the  lectures.  Some  minor  corrections  have  been  made. 
The  section  headings  have  been  inserted  over  the  sections. 


MATHEMATICAL  METHODS  IN  PHYSICS. 


199 


[Jan., 


This  edition  of  these  very  valuable  lectures  will  be  wel- 
comed. 

The  whole  field  of  functional  calculus  is  a new  territory  but 
recently  open  for  settlement,  though  an  adventurous  investi- 
gator occasionally  explored  small  parts  of  it  in  the  past  century. 
The  important  extensions  of  mathematics  have  come  from  the 
problems  of  inversion,  such  as  the  Galois  theory,  theory  of 
ideals,  differential  equations,  integral  equations,  and  now  the 
calcul  fonctionnel.  These  developments  of  Professor  Volterra 
are  of  the  highest  importance  mathematically  aside  from  all 
of  their  physical  interest,  for  the  reason  that  they  furnish 
a very  practical  path  of  entry  into  this  new  field  and  occupy 
a considerable  part  of  the  field  itself.  Fortunately  we  can 
follow  them  more  in  detail  in  the  two  recent  courses  of  his 
lectures,  Lemons  sur  les  fonctions  des  lignes,  and  Lemons  sur  les 
equations  integrates. 


James  Byrnie  Shaw. 


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